Recent questions in Advanced Math

Discrete mathAnswered question

excefebraxp 2022-09-07

Discrete math one to one question

Use the pigeonhole principle to prove that if $n\ge 2$ people go to the same party, there are at least 2 people who shake hands with the same number of other people.

Hint: Take the set of people at the party as your domain, define a function that evaluates how many people each person shook hands with.

Use the pigeonhole principle to prove that if $n\ge 2$ people go to the same party, there are at least 2 people who shake hands with the same number of other people.

Hint: Take the set of people at the party as your domain, define a function that evaluates how many people each person shook hands with.

Discrete mathAnswered question

peckishnz 2022-09-07

Proof by cases - discrete math

I need to prove by the cases technique that:

$If\text{}a,b\in Z,and\text{}a={b}^{3},\text{there is}\text{}c\in Z\text{}so\text{}a=9\cdot c\text{}or\text{}a=9\cdot c+1\text{}or\text{}a=9\cdot c-1.$

I need to prove by the cases technique that:

$If\text{}a,b\in Z,and\text{}a={b}^{3},\text{there is}\text{}c\in Z\text{}so\text{}a=9\cdot c\text{}or\text{}a=9\cdot c+1\text{}or\text{}a=9\cdot c-1.$

Upper Level MathAnswered question

moidu13x8 2022-09-07

Find the tight upper and lower bounds for the following recurrence:

$T(n)=2T\left(\frac{n}{2}\right)+6t\left(\frac{n}{3}\right)+{n}^{2},\text{}\text{if}\text{}n\ge 3=1\text{}\text{if}\text{}n\le 2$

$T(n)=2T\left(\frac{n}{2}\right)+6t\left(\frac{n}{3}\right)+{n}^{2},\text{}\text{if}\text{}n\ge 3=1\text{}\text{if}\text{}n\le 2$

Discrete mathAnswered question

Felix Cohen 2022-09-07

Discrete Math problem combinations and restrictions

I am trying to get my head around an idea but I can't seem to get it to work.

Imagine you have the word "MAMMAL"

Lets see I wanted to figure out how many ways I could rearrange the letters. Well that is easy. It is simply

$6!/3!2!=60$ possibilities

But what if I added a restriction to it such that all M's must be together.

Then I could consider all M's as one letter and it would be $4!/2!=12$ possibilities.

Again I understand that. But what if the restriction was there needs to be a minimum of 2 M's together at all times.

If I were to do the same process as the previous example, I would combine 2 M's into one. Which then would become $5!/2!=60$.

This seems to be a wrong answer because it is the same as my first calculation of finding all possibilities without any restrictions. Can anyone please explain to me as to how I need to approach the last problem of finding number of combinations where at least 2 M's are always together?

I am trying to get my head around an idea but I can't seem to get it to work.

Imagine you have the word "MAMMAL"

Lets see I wanted to figure out how many ways I could rearrange the letters. Well that is easy. It is simply

$6!/3!2!=60$ possibilities

But what if I added a restriction to it such that all M's must be together.

Then I could consider all M's as one letter and it would be $4!/2!=12$ possibilities.

Again I understand that. But what if the restriction was there needs to be a minimum of 2 M's together at all times.

If I were to do the same process as the previous example, I would combine 2 M's into one. Which then would become $5!/2!=60$.

This seems to be a wrong answer because it is the same as my first calculation of finding all possibilities without any restrictions. Can anyone please explain to me as to how I need to approach the last problem of finding number of combinations where at least 2 M's are always together?

Discrete mathAnswered question

Mutignaniz2 2022-09-07

Discrete Math Construct Tree from Weights

Consider the weights: 10, 12, 13, 16, 17, 17.

(a) Construct an optimal coding scheme for the weights given by using a tree.

(b) What is the total weight of the tree you found in part (a)?

Consider the weights: 10, 12, 13, 16, 17, 17.

(a) Construct an optimal coding scheme for the weights given by using a tree.

(b) What is the total weight of the tree you found in part (a)?

Discrete mathAnswered question

Staffangz 2022-09-07

Upperbound on harmonic number discrete math

How do I show ${H}_{{2}^{k}}\le k+1$ for each $k\ge 0$?

How do I show ${H}_{{2}^{k}}\le k+1$ for each $k\ge 0$?

Upper Level MathAnswered question

Dulce Cantrell 2022-09-07

I've been given the recursive function:

$$T(n)={n}^{\frac{2}{3}}\text{}T({n}^{\frac{1}{3}})+C$$

$$T(n<=2)=O(1)$$

With $n={2}^{{3}^{p}}$ where p is a positive integer.

$$T(n)={n}^{\frac{2}{3}}\text{}T({n}^{\frac{1}{3}})+C$$

$$T(n<=2)=O(1)$$

With $n={2}^{{3}^{p}}$ where p is a positive integer.

Discrete mathAnswered question

tamola7f 2022-09-07

Discrete Math

I am trying to understand a question I got:

Let U be a set.

For every $A\subseteq U$ we define the function: ${f}_{A}:P(U)\to P(U)$ such as fA(B)=(A∩B).

Prove: f is injective and surjective if and only if $A=U$

First, I do not understand what ${f}_{A}$ and ${f}_{A}(B)$ mean.

I think that if I will understand this, I will be able to start solving the question.

Any tips will be wonderful!

I am trying to understand a question I got:

Let U be a set.

For every $A\subseteq U$ we define the function: ${f}_{A}:P(U)\to P(U)$ such as fA(B)=(A∩B).

Prove: f is injective and surjective if and only if $A=U$

First, I do not understand what ${f}_{A}$ and ${f}_{A}(B)$ mean.

I think that if I will understand this, I will be able to start solving the question.

Any tips will be wonderful!

Discrete mathAnswered question

metal1fc 2022-09-07

Arranging books in bookshelves with the capacity of each shelf given

There are k identical bookshelves in which each shelf cannot contain m or more books. In how many ways can n distinct books be arranged on these k bookshelves?

If there is no condition on the capacity of each shelf, the number of ways to arrange books equals $\sum _{i\in [k]}L(n,i)$, where L(n,i) denotes the Lah number. However, because of that constraint, I have trouble in solving the problem.

I tried several ways to solve this problem by separating the cases via (1) the number of shelves which contains the full-number of books, or (2) the number of non-empty shelves. For the second trial, I observed that, if j denotes the number of non-empty shelves, then the number of ways to arrange the books is zero if $j<\lfloor n/m\rfloor $.

However, these methods does not proceed quite well, since it looks like these methods result in the recurrence relation rather than the exact form of the number.

For the related concepts, I have studied Catalan number, (both signed and unsigned) first and second Stirling number, Bell and Lah number, and the integer partition.

Any insight or comment are welcomed.

There are k identical bookshelves in which each shelf cannot contain m or more books. In how many ways can n distinct books be arranged on these k bookshelves?

If there is no condition on the capacity of each shelf, the number of ways to arrange books equals $\sum _{i\in [k]}L(n,i)$, where L(n,i) denotes the Lah number. However, because of that constraint, I have trouble in solving the problem.

I tried several ways to solve this problem by separating the cases via (1) the number of shelves which contains the full-number of books, or (2) the number of non-empty shelves. For the second trial, I observed that, if j denotes the number of non-empty shelves, then the number of ways to arrange the books is zero if $j<\lfloor n/m\rfloor $.

However, these methods does not proceed quite well, since it looks like these methods result in the recurrence relation rather than the exact form of the number.

For the related concepts, I have studied Catalan number, (both signed and unsigned) first and second Stirling number, Bell and Lah number, and the integer partition.

Any insight or comment are welcomed.

Discrete mathAnswered question

Gauge Odom 2022-09-07

How to prove that subset at odd size is equal to subset at even size?

I'm trying to prove that:

$$\sum _{i=0}^{\lceil \frac{n}{2}\rceil}{\textstyle (}\genfrac{}{}{0ex}{}{n}{2i}{\textstyle )}=\sum _{i=0}^{\lceil \frac{n}{2}\rceil}{\textstyle (}\genfrac{}{}{0ex}{}{n}{2i+1}{\textstyle )}$$

if n is odd, I can do it with the identity: $(}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}={\textstyle (}\genfrac{}{}{0ex}{}{n}{n-k}{\textstyle )$, because if we have odd number like 7 we have 4 ($=\lceil \frac{n}{2}\rceil $) pairs: (7,0),(6,1),(5,2),(4,3). The left element is odd and the right one is even. So:

$$(}\genfrac{}{}{0ex}{}{7}{0}{\textstyle )}={\textstyle (}\genfrac{}{}{0ex}{}{7}{7}{\textstyle )},\phantom{\rule{thinmathspace}{0ex}}{\textstyle (}\genfrac{}{}{0ex}{}{7}{6}{\textstyle )}={\textstyle (}\genfrac{}{}{0ex}{}{7}{1}{\textstyle )},...$$

But when I'm trying to do this about even numbers (like 8), I can't use this method.

So my question is: What can I do at cases of even numbers?

I'm trying to prove that:

$$\sum _{i=0}^{\lceil \frac{n}{2}\rceil}{\textstyle (}\genfrac{}{}{0ex}{}{n}{2i}{\textstyle )}=\sum _{i=0}^{\lceil \frac{n}{2}\rceil}{\textstyle (}\genfrac{}{}{0ex}{}{n}{2i+1}{\textstyle )}$$

if n is odd, I can do it with the identity: $(}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}={\textstyle (}\genfrac{}{}{0ex}{}{n}{n-k}{\textstyle )$, because if we have odd number like 7 we have 4 ($=\lceil \frac{n}{2}\rceil $) pairs: (7,0),(6,1),(5,2),(4,3). The left element is odd and the right one is even. So:

$$(}\genfrac{}{}{0ex}{}{7}{0}{\textstyle )}={\textstyle (}\genfrac{}{}{0ex}{}{7}{7}{\textstyle )},\phantom{\rule{thinmathspace}{0ex}}{\textstyle (}\genfrac{}{}{0ex}{}{7}{6}{\textstyle )}={\textstyle (}\genfrac{}{}{0ex}{}{7}{1}{\textstyle )},...$$

But when I'm trying to do this about even numbers (like 8), I can't use this method.

So my question is: What can I do at cases of even numbers?

Discrete mathAnswered question

Jimena Hatfield 2022-09-07

A tight bound for $T(n)={2}^{n}T(\frac{n}{2})+{n}^{n}$

Given recurrence

$$T(n)={2}^{n}T\left(\frac{n}{2}\right)+{n}^{n}$$

How we can show that $T(n)\le {n}^{n}$?

I show that $T(n)\ge {n}^{n}$ because of existence the term ${n}^{n}$ in T(n).

Given recurrence

$$T(n)={2}^{n}T\left(\frac{n}{2}\right)+{n}^{n}$$

How we can show that $T(n)\le {n}^{n}$?

I show that $T(n)\ge {n}^{n}$ because of existence the term ${n}^{n}$ in T(n).

Upper Level MathAnswered question

Alfredeim 2022-09-07

a is a factor of 10 and b is a factor of 15.

$\begin{array}{|cc|}\hline \text{Column A}& \text{Column B}\\ \text{The smallest value that ab must be a factor of}& 30\\ \hline\end{array}$

Is Column A greater than/ smaller than / equal to Column B?

$\begin{array}{|cc|}\hline \text{Column A}& \text{Column B}\\ \text{The smallest value that ab must be a factor of}& 30\\ \hline\end{array}$

Is Column A greater than/ smaller than / equal to Column B?

Discrete mathAnswered question

Jazmyn Saunders 2022-09-07

Help with summations for discrete math

I could start them but I have no idea how to simplify after expanding them.

$$\sum _{i=28}^{n}(3{i}^{2}-4i+(5/{7}^{i}))$$

$$\sum _{i=1}^{n}\frac{i}{{2}^{n-i+1}}$$

I could start them but I have no idea how to simplify after expanding them.

$$\sum _{i=28}^{n}(3{i}^{2}-4i+(5/{7}^{i}))$$

$$\sum _{i=1}^{n}\frac{i}{{2}^{n-i+1}}$$

Upper Level MathAnswered question

mashingcho9v 2022-09-07

I have an upper and lower bound number:

upper: 21

lower: 3

I then have a second number that can be anywhere between this range, I would like the second number to increment faster when it is closer to the lower bound and slower when it is closer to the upper bound.

How can I achieve this using mathematics?

upper: 21

lower: 3

I then have a second number that can be anywhere between this range, I would like the second number to increment faster when it is closer to the lower bound and slower when it is closer to the upper bound.

How can I achieve this using mathematics?

Discrete mathAnswered question

Zackary Duffy 2022-09-07

How can i prove this?

$\mathrm{\forall}n\mathrm{\exists}p({p}^{2}\le n<(p+1{)}^{2})$

The domain of quantifiers is N.

$\mathrm{\forall}n\mathrm{\exists}p({p}^{2}\le n<(p+1{)}^{2})$

The domain of quantifiers is N.

Discrete mathAnswered question

ezelsbankuk 2022-09-07

What is a strongly discrete sequence?

What is the definition (and possibly examples or applications) of strongly discrete sequence? I have seen this term, but I am not able to find anything about it.

What is the definition (and possibly examples or applications) of strongly discrete sequence? I have seen this term, but I am not able to find anything about it.

Discrete mathAnswered question

Ciolan3u 2022-09-07

Discrete math De Morgan's Laws. So like if I have a form ${p}^{q}$ and say I make p equal giving you cookies and q giving you milk so the sentence is just "Giving you cookies and giving you milk" like when I think about it if I think of the opposite(negation) I just automatically think it's not giving you cookies and not giving you milk but this is wrong but is it really? I mean am I just misunderstanding what opposite/negation means here?

Discrete mathAnswered question

Kailey Vargas 2022-09-07

Using Direct Proofs in Discrete Math

Use a direct proof to show that if x is a rational number, then ${x}^{2}$ is also a rational number.

I know I need to use the definition of rational numbers but don't know how to do it in this problem.

Use a direct proof to show that if x is a rational number, then ${x}^{2}$ is also a rational number.

I know I need to use the definition of rational numbers but don't know how to do it in this problem.

Discrete mathAnswered question

atarentspe 2022-09-07

Validity of arguments in discrete math

All girls are tall.

Anyone who is tall and dark will pass.

Claire is a girl.

Conclusion: Claire will pass.

converting the statements to predicate logic:

- $G(x)=x$ is a girl

- $A(x)=x$ is tall

- $B(X)=x$ is dark

- $C(x)=x$ passes

$\phantom{\rule{1em}{0ex}}\forall (x)[G(x)\to A(x)]\phantom{\rule{0ex}{0ex}}\forall (x)[[A(x)\wedge B(x)]\to C(x)]\phantom{\rule{0ex}{0ex}}G(c)\to A(c)$ .....................c stands for claire

Am i right till now? and please help me proceed to finish the problem .

All girls are tall.

Anyone who is tall and dark will pass.

Claire is a girl.

Conclusion: Claire will pass.

converting the statements to predicate logic:

- $G(x)=x$ is a girl

- $A(x)=x$ is tall

- $B(X)=x$ is dark

- $C(x)=x$ passes

$\phantom{\rule{1em}{0ex}}\forall (x)[G(x)\to A(x)]\phantom{\rule{0ex}{0ex}}\forall (x)[[A(x)\wedge B(x)]\to C(x)]\phantom{\rule{0ex}{0ex}}G(c)\to A(c)$ .....................c stands for claire

Am i right till now? and please help me proceed to finish the problem .

Discrete mathAnswered question

Ciolan3u 2022-09-07

Assuming D is the set of all dogs, H is the set of all homes, and BT(d,h) means that the dog d is in home h

Let's say I have a group of dogs and I want to say that every dog is in exactly one home. How would I represent that using predicate logic? So far I got, For all d in D, There exists a h in H, belongs to (d,H) AND....

Let's say I have a group of dogs and I want to say that every dog is in exactly one home. How would I represent that using predicate logic? So far I got, For all d in D, There exists a h in H, belongs to (d,H) AND....

Students pursuing advanced Math are constantly dealing with advanced Math equations that are mostly used in space engineering, programming, and construction of AI-based solutions that we can see daily as we are turning to automation that helps us to find the answers to our challenges. If it sounds overly complex with subjects like exponential growth and decay, don’t let advanced math problems frighten you because these must be approached through the lens of advanced Math questions and answers. Regardless if you are dealing with simple equations or more complex ones, just break things down into several chunks as it will help you to find the answers.